What weights should I use while back-propagating?
First I will assume you notate $y$ as the models output and $z$ as the ground-truth. Second, I am assuming this is a linear model (No activation functions). Then the gradient math goes as so:

$
\begin{align*}
\frac{dE}{dw_{ij}^1} &amp;= \frac{dE}{dy}\frac{dy}{dw_{ij}^1} \\
&amp;= \frac{dE}{dy}\sum_k\frac{\partial y}{\partial n_{k}^3}\frac{dn_{k}^3}{dw_{ij}^1} \\
&amp;= \frac{dE}{dy}\sum_k\frac{\partial y}{\partial n_{k}^3} \frac{\partial n_{k}^3}{ \partial n_{j}^2} \frac{dn_{j}^2}{ dw_{ij}^1}  \\
&amp;= -2*(z-y)\sum_kw_{kl}^3w_{jk}^2 x_i  \\ 
\end{align*}
$  

So the reason you are having trouble to figure out which $k$ index to use, the answer is because you need to use both and sum over them. The $l$ index is just the only $l$ index that exists because you only have one node in that layer.